Single-layer networks are often represented by an adjacency matrix, which is helpful for
extracting information about the network by evaluating matrix functions or by computing
certain eigenvectors. For instance, Estrada and Higham [1]} describe how the matrix
exponential and resolvent can be used to determine how easy it is to communicate between
nodes in a single-layer network, and which nodes are the most important ones; see also
Estrada [2]} and references therein. For multilayer networks, we use tensors,
i.e., a multidimensional generalization of matrices, and represent the network by a
\(2(d+1)\) -order adjacency tensor \(\mathcal {A}\)  of size
\((|V|\times |L_1|\times \ldots \times |L_d|)\times (|V|\times |L_1|\times \ldots \times |L_d|)\) ,
where \(|V|\)  denotes the number of nodes in each layer, and \(|L_j|\) , with
\(j=1,2,\ldots ,d\) , designates the number of layers. The entry
\(\mathcal {A}(i,\ell _1,\ldots ,\ell _d,j,k_1,\ldots ,k_d)\)  of the tensor \(\mathcal {A}\)  for an unweighted
network with layers \(L_j\) , \(j=1,2,\ldots ,d\)  is one if there is an edge
\(v_i^{\ell _1,\ldots ,\ell _d}\rightarrow v_j^{k_1,\ldots ,k_d}\) ; otherwise the tensor
entry is zero. For a weighted network the entry entry is replaced by a real,
generally positive, number. In a directed network some of the edges represent “one-way
streets”.
